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Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
In probability theory and related fields, a stochastic (/ s t ə ˈ k æ s t ɪ k /) or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time.
[27] [28] Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory". [28] In mathematics, the theory of stochastic processes is an important contribution to probability theory, [29] and continues to be an active topic of research for both theory and applications. [30] [31] [32]
A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness .
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, [1] is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability =.
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of ...
Probability is used to design games of chance so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play. [26] Another significant application of probability theory in everyday life is reliability.
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.